For which value of x horizontal Tangent Line

In summary, to find the value of x that results in a horizontal tangent line for the function f(x) = k/(ax^2 + bx + c), you can use the quotient rule to set the derivative equal to 0 and solve for x. However, there are certain restrictions on the values of a, b, and c. Alternatively, if the numerator is just a constant, you can directly use the quotient rule in a shorter format.
  • #1
chaosblack
16
0
For which value of x...horizontal Tangent Line

Homework Statement



For which value of x does f(x) = [tex]\frac{k}{ax^{2}+bx+c}[/tex] have a horizontal tangent line?


Homework Equations



Quotient Rule?

F'(x) = [g(x)a'(x) - a(x)g'(x)]/g(x)^2?


The Attempt at a Solution



Am I supposed to just sub it into a quotient rule format, making the derivative equal to 0?

So it would look like

0 = 0 - (2ax + b)/[[ax[tex]^{2}[/tex]+bx+c][tex]^{2}[/tex]]? (and then simplify of course?)
 
Last edited:
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  • #2
That's correct. But there are certain restrictions on the values of a, b, and c.

(Also, when there is only a constant in the numerator, like f(x) = k/g(x), then you can directly use f'(x) = k d/dx[1/(g(x)] = k[-g'(x)/[g(x)^2], which is nothing but the quotient rule in a lesser number of steps.)
 
  • #3
okay thanks, so the answer would just be the derivative set equal to 0?
 
  • #4
yes it is
 
  • #5
okay thanks alot
 

1. What is a horizontal tangent line?

A horizontal tangent line is a line that touches a curve at a single point and is parallel to the x-axis at that point. This means that the slope of the tangent line is 0 at that point.

2. How do you find the value of x for a horizontal tangent line?

The value of x for a horizontal tangent line can be found by setting the derivative of the function equal to 0 and solving for x. This will give you the x-coordinate of the point where the tangent line is horizontal.

3. Why is finding the value of x for a horizontal tangent line important?

Finding the value of x for a horizontal tangent line is important because it helps to identify critical points on a curve. These points can give valuable information about the behavior of the function and can be used to find local extrema.

4. Can a function have more than one value of x for a horizontal tangent line?

Yes, a function can have multiple values of x for a horizontal tangent line. This can happen when the function has a flat region or a plateau where the slope is 0 at multiple points.

5. How does the graph of a function change when there is a horizontal tangent line?

When there is a horizontal tangent line on a graph, it means that the slope of the function is 0 at that point. This can result in a flat region or a plateau on the graph. The function may also have a local maximum or minimum at that point.

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